If a cube number by multiplying any number make a cube number, the multiplied number will also be cube. For let the cube number A by multiplying any number B make the cube number C; I say that B is cube. For let A by multiplying itself make D; therefore D is cube. [IX. 3] Now, since A by multiplying itself has made D, and by multiplying B has made C, therefore, as A is to B, so is D to C. [VII. 17] And since D, C are cube, they are similar solid numbers. Therefore two mean proportional numbers fall between D, C. [VIII. 19] And, as D is to C, so is A to B; therefore two mean proportional numbers fall between A, B also. [VIII. 8]